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If you represent a second-order polynomial $s(x)$ with Lagrange polynomials $L_i(x)$ and interpolation points $\beta_i$, $i=0,1,2$, such that $$s(x)=s(\beta_0)L_0(x)+s(\beta_1)L_1(x)+s(\beta_2)L_2(x)\tag{1}$$ then for the equality in $(1)$ to be satisfied, the polynomials $L_i(x)$ must have zeros at $x=\beta_j$, $j\neq i$, and they must equal $1$ at $x=\...


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